Unique real‐variable expressions of displacement and traction fundamental solutions covering all transversely isotropic elastic materials for 3D BEM

Abstract

AbstractA general, efficient and robust boundary element method (BEM) formulation for the numerical solution of three‐dimensional linear elastic problems in transversely isotropic solids is developed in the present work. The BEM formulation is based on the closed‐form real‐variable expressions of the fundamental solution in displacements Uik and in tractions Tik, originated by a unit point force, valid for any combination of material properties and for any orientation of the radius vector between the source and field points. A compact expression of this kind for Uik was introduced by Ting and Lee (Q. J. Mech. Appl. Math. 1997; 50:407–426) in terms of the Stroh eigenvalues on the oblique plane normal to the radius vector. Working from this expression of Uik, and after a revision of their final formula, a new approach (based on the application of the rotational symmetry of the material) for deducing the derivative kernel Uik, j and the corresponding stress kernel Σijk and traction kernel Tik has been developed in the present work. These expressions of Uik, Uik, j, Σijk and Tik do not suffer from the difficulties of some previous expressions, obtained by other authors in different ways, with complex‐valued functions appearing for some combinations of material parameters and/or with division by zero for the radius vector at the rotational‐symmetry axis. The expressions of Uik, Uik, j, Σijk and Tik have been presented in a form suitable for an efficient computational implementation. The correctness of these expressions and of their implementation in a three‐dimensional collocational BEM code has been tested numerically by solving problems with known analytical solutions for different classes of transversely isotropic materials. Copyright © 2007 John Wiley & Sons, Ltd.